Math Unit 1

CCGPS
Frameworks

Mathematics

Kindergarten
Unit 1: Counting With Friends

Georgia Department of Education
Common Core Georgia Performance Standards Framework
Kindergarten Mathematics Unit 1

Unit 1: Counting With Friends (5 weeks)

TABLE OF CONTENTS

Overview ............................................................................................................................................

Key Standards and Related Standards ........................................................................

Number Sense Trajectory ..................................................................................................................

Enduring Understandings...................................................................................................................

Essential Questions ............................................................................................................................

Concepts and Skills to Maintain ........................................................................................................

Select Terms and Symbols .................................................................................................................

Strategies for Teaching and Learning ................................................................................................

Common Misconceptions ..................................................................................................................

Evidence of Learning .........................................................................................................................

Tasks ..................................................................................................................................................

Got Dots? (0-10)
Numerals, Pictures, Words (0-10)
Fill the Line (0-9)
What the Heck is Rekenrek?
Fill the Chutes
Race to 20
The Cardinal Cup (0-10)
Order the Dice
More or Less
How Many Are in the Bag?
More or Less-Make a Guess
Find the 5th Tower

MATHEMATICS  GRADE K UNIT 1: Counting With Friends
Dr. John D. Barge, State School Superintendent
April 2012  Page 2 of 66
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The Critical Areas are designed to bring focus to the standards at each grade by describing
the big ideas that educators can use to build their curriculum and to guide instruction.

Students use numbers, including written numerals, to represent quantities and to solve
quantitative problems, such as counting objects in a set; counting out a given number of objects;
comparing sets or numerals; and modeling simple joining and separating situations with sets of
objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students
should see addition and subtraction equations, and student writing of equations in kindergarten is
encouraged, but it is not required.) Students choose, combine, and apply effective strategies for
answering quantitative questions, including quickly recognizing the cardinalities of small sets of
objects, counting and producing sets of given sizes, counting the number of objects in combined
sets, or counting the number of objects that remain in a set after some are taken away.

OVERVIEW
In this unit, students will start kindergarten thinking of counting as a string of words, but then
they make a gradual transition to using counting as a tool for describing their world. They must
construct the idea of counting using manipulatives and other resources to see the numbers
visually (dot cards, tens frames). To count successfully, students must remember the rote
counting sequence, assign one counting number to each object counted, and at the same time
have a strategy for keeping track of what has already been counted and what still needs to be
counted. Only the counting sequence is a rote procedure. The meaning students attach to
counting is the key conceptual idea on which all other number concepts are developed. Students
will develop successful and meaningful counting strategies as they practice counting and as they
listen to and watch others count.

Although the units in this instructional framework emphasize key standards and big ideas at
specific times of the year, routine topics such as counting, positional word, time should be
addressed on an ongoing basis through the use of calendar, centers (tubs), and games. This unit
should allow students to understand the concepts of numbers and counting.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important
that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A
variety of resources should be utilized to supplement this unit. This unit provides much needed
content information as well as excellent learning activities. The task in this unit illustrates the
types of learning activities that should be utilized from a variety of sources.

The Critical Areas are designed to bring focus to the standards at each grade by describing the
big ideas that educators can use to build their curriculum and to guide instruction. In
Kindergarten, instructional time should focus on two critical areas: (1) representing, relating, and
operating on whole numbers, initially with sets of objects. Students choose, combine, and apply
effective strategies for answering quantitative questions, including quickly recognizing the
cardinalities of small sets of objects, counting and producing sets of given sizes, counting the
number of objects in combined sets, or counting the number of objects that remain in a set after
some are taken away. Students use numbers, including written numerals, to represent quantities
and to solve quantitative problems, such as counting objects in a set; counting out a given number
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of objects; comparing sets or numerals; and modeling simple joining and separating situations
with sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten
students should see addition and subtraction equations, and student writing of equations in
kindergarten is encouraged, but it is not required.)

STANDARDS FOR MATHEMATICAL CONTENT

Counting and Cardinality

Know number names and the count sequence.

MCC.K.CC.1. Count to 100 by ones and by tens.

MCC.K.CC.2. Count forward beginning from a given number within the known sequence
(instead of having to begin at 1).

MCC.K.CC.3. Write numbers from 0 to 20. Represent a number of objects with a written
numeral 0-20 (with 0 representing a count of no objects).

Count to tell the number of objects.

MCC.K.CC.4. Understand the relationship between numbers and quantities; connect
counting to cardinality.
a. When counting objects, say the number names in the standard order,
pairing each object with one and only one number name and each
number name with one and only one object.
b. Understand that the last number name said tells the number of objects
counted. The number of objects is the same regardless of their
arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity that
is one larger.

Classify objects and count the number of objects in each category.

MCC.K.MD.3. Classify objects into given categories; count the numbers of objects in each
category and sort the categories by count.

(For descriptors of standard cluster please see the Grade Level Overview)

STANDARDS FOR MATHEMATICAL PRACTICE

The standards for mathematical practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students. These practices rest on important
“processes and proficiencies” with longstanding importance in mathematics education.

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Students are expected to:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

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Number Sense Trajectory –Putting It All Together
Subitizing Comparison Counting One-to-One Cardinality Hierarchical Inclusion Number
Being able to Being able to Rote procedure of Correspondence Tells how many things are Numbers are nested Conservation
Trajectory

visually compare counting. The Students can in a set. When counting a inside of each other and The number of
recognize a quantities by meaning attached connect one set of objects, the last word that the number grows objects remains the
quantity of 5 identifying to counting is number with one in the counting sequence by one each count. 9 is same when they are
or less. which has more developed through object and then names the quantity for that inside 10 or 10 is the rearranged spatially.
and which has one-to-one count them with set. same as 9 + 1. 5 is 4&1 OR 3&2.
less. correspondence. understanding.

Each concept builds on the previous idea and students should explore and construct concepts in such a sequence

Spatial Relationship One and Two-More or Less Understanding Anchors Part-Part-Whole
Patterned Set Recognition Students need to understand the relationship of number as it relates to Students need to see the Relationship
Relationships

Students can learn to recognize +/- one or two. Here students should begin to see that 5 is 1 more relationship between Students begin to
Number

sets of objects in patterned than 4 and that it is also 2 less than 7. numbers and how they conceptualize a
arrangements and tell how many relate to 5s and 10s. 3 number as being
without counting. is 2 away from 5 and 7 made up from two or
away from 10. more parts.

Addition and Subtraction Strategies
One/Two More/Less Make a Ten Near Doubles
These facts are a direct Use a quantity from Using the doubles
application of the One/Two one addend to give to anchor and
More/ Less than another to make a ten combining it with 1
relationships then add the remainder. and 2 more/less.
9 + 7 = 10 + 6
Facts with Zero Doubles
Need to be introduced so Many times students
that students don’t will use doubles as an
overgeneralize that answers anchor when adding
to addition are always and subtracting.
bigger.

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ENDURING UNDERSTANDINGS

Number Properties

Understand numbers, ways of representing numbers, relationships among numbers, and
number systems.
Count with understanding and recognize “how many” in a set of objects.
Develop a sense of whole numbers and represent and use them in flexible ways.
Develop understanding of the relative magnitude and position of whole numbers.
Use multiple models to develop initial understandings of the base-ten number system.
Connect number words and numerals to the quantities they represent, using various
physical models and representation
Counting tells how many things are in a set.
The last number word, when counting, names the quantity in a set.
A number can be represented by a set of objects, then by a word, and finally by a
numeral.
Numbers are related to each other through a variety of relationships. For example, 6 is
one more than 5, and is 4 less than 10.
Counting can be a way to gather information.

Coins are not explicitly taught in kindergarten, but the connections to patterns and skip
counting should be made. Coins can be used as a manipulative for patterns, skip counting
and counting.

ESSENTIAL QUESTIONS

How can numbers be represented?
How can playing board games make me a better mathematician?
How can we record what we count?
How can we show numbers in different ways?
How can you know an amount without counting each object?
How do we know if a number is more or less than another number?
How do we use counting in our everyday life?
How do we use numbers every day?
How does putting things in order keep things organized?
How many ways can I group objects using cardinal and ordinal numbers?
What do numbers mean to us?
What is a numeral?
What is the difference between “more” and “less”?
What types of questions should I ask myself or my partner when playing a math game?
Why are numbers important?
Why do we need to be able to count forwards and backwards?
Why do we need to be able to count objects?
Why do we need to be able to put things in order?
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Why do we need to be able to read ordinal numbers?
Why is it important to know how to put things in number order?
Why would we need to be able to read number words?

CONCEPTS/SKILLS TO MAINTAIN

Although many students may have attended pre-school prior to entering kindergarten, this is the
first year of school for some students. For that reason, no concepts/skills to maintain will be
listed at this time. It is expected that teachers will differentiate to accommodate those students
that may enter kindergarten with prior knowledge.

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive
list and should not be taught in isolation. However, due to evidence of frequent difficulty and
misunderstanding associated with these concepts, instructors should pay particular attention to
them and how their students are able to explain and apply them.

Teachers should first present these concepts to students with models and real life examples.
Students should understand the concepts involved and be able to recognize and/or use them with
words, models, pictures, or numbers.

Zero
Order
Number Line
Forward
Backward
Count
Counting-On
Compare
Digits
Number
Numeral
Less than
More than/Greater than
Model
Number
Numeral
Ones
Pair
Quantity
Same
Sequence
Set

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STRATEGIES FOR TEACHING AND LEARNING

Provide settings that connect mathematical language and symbols to the everyday lives of
kindergarteners. Support students’ ability to make meaning and mathematize the real world. Help
them see patterns, make connections and provide repeated experiences that give students time
and opportunities to develop understandings and increase fluency. Encourage students to explain
their reasoning by asking probing questions, such as, “How do you know?"
Students view counting as a mechanism used to land on a number. Young students mimic
counting often with initial lack of purpose or meaning. Coordinating the number words, touching
or moving objects in a one-to-one correspondence may be little more than a matching activity.
However, saying number words as a chant or a rote procedure plays a part in students
constructing meaning for the conceptual idea of counting. They will learn how to count before
they understand cardinality, i.e. that the last count word is the amount of the set.
Counting on or counting from a given number conflicts with the learned strategy of counting
from the beginning. In order to be successful in counting on, students must understand
cardinality. Students often merge or separate two groups of objects and then re-count from the
beginning to determine the final number of objects represented. For these students, counting is
still a rote skill or the benefits of counting on have not been realized. Games that require students
to add on to a previous count to reach a goal number encourage developing this concept.
Frequent and brief opportunities utilizing counting on and counting back are recommended.
These concepts emerge over time and cannot be forced.
Like counting to 100 by either ones or tens, writing numbers from 0 to 20 is a rote process.
Initially, students mimic the actual formation of the written numerals while also assigning it a
name. Over time, children create the understanding that number symbols signify the meaning of
counting. Numerals are used to communicate across cultures and through time a certain meaning.
Numbers have meaning when children can see mental images of the number symbols and use
those images with which to think. Practice count words and written numerals paired with
pictures, representations of objects, and objects that represent quantities within the context of life
experiences for kindergarteners. For example, dot cards, dominoes and number cubes all create
different mental images for relating quantity to number words and numerals.
One way students can learn the left to right orientation of numbers is to use a finger to write
numbers in air (sky writing). Children will see mathematics as something that is alive and that
they are involved.
Students should study and write numbers 0 to 20 in this order: numbers 1 to 9, the number 0, and
then numbers 10 to 20. They need to know that 0 is the number items left after all items in a set
are taken away. Do not accept “none” as the answer to “How many items are left?” for this
situation.

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One of the first major concepts in a student’s mathematical development is cardinality.
Cardinality, knowing that the number word said tells the quantity you have and that the number
you end on when counting represents the entire amount counted. The big idea is that number
means amount and, no matter how you arrange and rearrange the items, the amount is the same.
Until this concept is developed, counting is merely a routine procedure done when a number is
needed. To determine if students have the cardinality rule, listen to their responses when you
discuss counting tasks with them. For example, ask, “How many are here?”. The student counts
correctly and says that there are seven. Then ask, “Are there seven?”. Students may count or
hesitate if they have not developed cardinality. Students with cardinality may emphasize the last
count or explain that there are seven because they counted them. These students can now use
counting to find a matching set.
Students develop the understanding of counting and cardinality from experience. Almost any
activity or game that engages children in counting and comparing quantities, such as board
games, will encourage the development of cardinality. Frequent opportunities to use and discuss
counting as a means of solving problems relevant to kindergarteners, is more beneficial than
repeating the same routine day after day. For example, ask students questions that can be
answered by counting up to 20 items before they change, and as they change, locations
throughout the school building.
As students develop meaning for numerals, they also compare numerals to the quantities they
represent. Models that can represent numbers – such as dot cards and dominoes – become tools
for such comparisons. Students can concretely, pictorially or mentally look for similarities and
differences in the representations of numbers. They begin to “see” the relationship of one more,
one less, two more and two less, thus landing on the concept that successive numbers name
quantities that are one larger. In order to encourage this idea, children need discussion and
reflection of pairs of numbers from 1 to 10. Activities that utilize anchors of 5 and 10 are helpful
in securing understanding of the relationships between numbers. This flexibility with numbers
will build students’ ability to break numbers into parts.
Provide a variety of experiences in which students connect count words or number words to the
numerals that represent the quantities. Students will arrive at an understanding of a number when
they acquire cardinality and can connect a number with the numerals and the number word for
the quantity they all represent.

Special Note:
Although the standard MCC.K.CC.1. (Count to 100 by ones and by tens) is included throughout
this unit, students should be given ample time to count and really focus on numbers through 20.
This standard is seen as a progression that is to be met by the end of the year. Although the
standard states “to 100”, this unit (in particular the tasks), focus on numbers and number
relationships through 10. It is because of the aforementioned reasons that skip counting is not
specifically addressed in this unit.

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COMMON MISCONCEPTIONS
Some students might not see zero as a number. Ask students to write 0 and say zero to
represent the number of items left when all items have been taken away. Avoid using the
word none to represent this situation.
Some students might think that the count word used to tag an item is permanently connected to
that item. So when the item is used again for counting and should be tagged with a different
count word, the student uses the original count word. For example, a student counts four
geometric figures: triangle, square, circle and rectangle with the count words: one, two, three,
and four. If these items are rearranged as rectangle, triangle, circle and square and counted, the
student says these count words: four, one, three, and two.

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following
competencies:
begin to count by ones and tens to 100 (rote count)
continue count sequence when beginning from a number greater than 1
subitize quantity up to 5
count objects to 20
sequence numbers to 20
one to one correspondence
identify a number quantity for numerals and words
represent numbers with numerals, pictures and words
understand numbers and the relationships between quantities
understand positional and ordinal words
understand the concept of “more” and “less”

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TASKS
Scaffolding Task Constructing Task Practice Task Performance Tasks
Constructing
Tasks that build up Summative
understanding through
to the constructing Games/activities assessment for the
deep/rich contextualized
task unit
problem solving tasks

Task Name Task Type/ Grouping Strategy Content Addressed

Scaffolding Subitizing, Counting objects to 10,
Got Dots? (0-10)
Whole/Small/Partner/Individual Sequencing Numbers,
Subitizing, Counting objects to 10,
Numerals, Pictures, Constructing Task
Sequencing Numbers, Matching
Words (0-10) Whole/Small/Partner/Individual
Number Words to Numbers
Constructing Task Numeral recognition, number word
Fill the Line (0-9)
Whole/partner recognition, Numeral writing
What the Heck is Constructing Task Subitizing, Modeling numbers,
Rekenrek? Partner Understanding number relationships
Practice Task
Fill the Chutes One to one correspondence
Whole/Small/Partner/Individual
Practice Task
Race to 20 One to one correspondence
Partner
Constructing Task One to one correspondence, Counting
The Cardinal Cup (0-10)
Whole/Partner objects to 10, Numeral recognition
Constructing Task Sequencing Numerals, numeral
Order the Dice Partner recognition
Constructing Task Numeral recognition, Understanding
More or Less Whole/Partner number relationships
How Many Are in the Constructing Task Numeral recognition, Understanding
Bag? Small groups, individual number relationships
Numeral recognition, Sequencing
More or Less-Make a Practice Task
Numerals, Understanding number
Guess Whole/Partner
relationships
One to one correspondence,
Performance Task Understanding number relationships,
Find the 5th Tower
Whole group/Small group Numeral writing, Positional words,
Ordinal words

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SCAFFOLDING TASK: Got Dots? (0-10)
This task contains numerous activities where students can engage using
different the different representations of numbers. (Adapted from VdW Backline Masters)


MCC.K.CC.1 Count to 100 by ones and by tens.



MCC.K.CC.4. Understand the relationship between numbers and quantities; connect counting to
cardinality.
a. When counting objects, say the number names in the standard order, pairing each object
with one and only one number name and each number name with one and only one
object.
b. Understand that the last number name said tells the number of objects counted. The
number of objects is the same regardless of their arrangement or the order in which they
were counted.
c. Understand that each successive number name refers to a quantity that is one larger.



BACKGROUND KNOWLEDGE

This task contains numerous activities where students engage in subitizing activities.
Subitizing introduces basic ideas of cardinality- “how many”, ideas of “more” and “less,” ideas
of parts and wholes and their relationships, beginning arithmetic, and, in general, ideas of
quantity. Developed well, these are related, forming webs of connected ideas that are the
building blocks of mathematics through elementary, middle, and high school, and beyond.
(Clementes & Sarama, Learning and Teaching Early Math, 2009)

The subitizing of quantities can be achieved with dot cards, ten frames, and base-ten
manipulatives later on. Using recognizable patterns like the ones found on dice are patterns that
are instantly recognizable to most kindergarten students to game play. Many of the tasks

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included throughout this unit involving subitizing and dot cards should be continued throughout
the year.

ESSENTIAL QUESTIONS

How can you know an amount without counting each object?

MATERIALS

Dot cards (recommend printing multiple sets of cards on tag board and laminating)

GROUPING

Whole group and partner task

TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

Dot Flash: Teacher/Student flashes a dot card to class/partner and quickly covers it up.
Students must say the quantity of dots they saw and describe how they know what they
saw. Example: I saw 4 dots because I saw a group of 3 dots and there was one left over
to make 4.The difficulty in the game can be increased by the amount of time that the dots
are shown to students.

Count ‘Em: a card is turned over. The first player to say the quantity of dots on the
cards keeps that card. Partner must count the dots on the card to verify. No assuming

One More/Less: same as dot flash but students need to say either 1 more or less than the
dots on the card. Whether it is more or less must be established before the game begins.

Who Has More/Less/Same?: 2 players turn over 1 card at the same time. first player to
identify which card has more/less/same keeps the 2 cards.

Line ‘Em Up: give a student a set of cards and have them line the cards up in a specific
order. (least to greatest – forward counting sequence, greatest to least-backward counting
sequence)

Kindergarten students are extremely creative and continuously invent new games. Have students
create a game using the cards and share with classmates. Van de Walle’s Teaching Student
Centered Mathematics k-3, lists numerous ways to incorporate subitizing activities into the
classroom. A greater variety of dot cards and dot plates can be found online and Van de Walle’s
Blackline Masters Series at http://www.ablongman.com/vandewalleseries/volume_1.html. In
addition Van de Walle suggests numerous ways that activities and tasks can be repeated
throughout the school year as centers or stations.
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FORMATIVE ASSESSMENT QUESTIONS

How do you know that you counted correctly?
How many dots did you see?
How do you know?
What way did you see the dots grouped together?
How many dots away from 5 is 8? How many dots would you need to make 10?
(anchoring 5&10)

DIFFERENTIATION

Extension and Intervention
Increasing or decreasing the quantity of dots on a card can help with differentiating
subitizing

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CONSTRUCTING TASK: Numerals-Pictures-Words
This task contains numerous activities where students can engage
using different the different representations of numbers.



cardinality.
object.
were counted.




Students need to understand that quantity can be represented through numerals, pictures, and
words. Students should be given ample time to explore this concept early on in kindergarten.
These task cards are designed for students to see and recognize the different forms in which a
quantity can be represented.
ESSENTIAL QUESTIONS

How do we know if a number is more or less than another number?

MATERIALS

Numerals, Pictures, Words playing cards

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GROUPING

Whole group, small group, partner, individual


Concentration/Memory: Shuffle the cards and lay them face down in a pattern. Let students
decide the pattern but they need to explain their patter. On each turn, a player turns over two
cards (one at a time) and keeps them if they match numbers. If they successfully match a pair of
numbers, that player also gets to take another turn. When a player turns over two cards that do
not match numbers, those cards are turned face down again and it becomes the next player’s turn.
Players keep each pair they find. At the end of the game, each pair scores one point. When all the
pairs have been found, the player with the most points wins.

Squeeze: Cards are placed face down in a stack on the table. The first player takes two cards and
places them face up on the table with a space between them and in order from smallest to largest.
The second player does the same. They then turn up the top card in the pile. If this card squeezes
between the two cards that player gets a point. If Payer 1 has “2” and “5” and Player 2 has “4”
and “9” and a “3” is flipped over, only Player 1 gets a point because “3” fits between their
numbers. Keep score on a ten-frame. First player to 10 wins.

Got Dots: The subitizing activities listed in Got Dots can also be included and played with the
Numerals, Pictures,Words Cards.


How many dots did you see?
How do you know?
What way did you see the dots grouped together?
How many dots is 8 from 5? How many dots would you need to make 10? (anchoring
5&10)

DIFFERENTIATION

Extension and Intervention

Increasing or decreasing the quantity of dots on a card can help with differentiating
subitizing activities.

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0 zero

1 one
●

2 two
● ●

3 three
● ● ●

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4 four
● ● ● ●

5 five
● ● ● ● ●

6 six
● ● ● ● ●
●

7 seven
● ● ● ● ●
● ●

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8 eight
● ● ● ● ●
● ● ●

9 nine
● ● ● ● ●
● ● ● ●

10 ten
● ● ● ● ●
● ● ● ● ●

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CONSTRUCTING TASK: Fill in the Line 0 to 9
Approximately 1 Day repeated as a station (Adapted from Race to Trace from
www.K-5TeacherResources.com





cardinality.
a) When counting objects, say the number names in the standard order, pairing each object
object.
b) Understand that the last number name said tells the number of objects counted. The
were counted.
c) Understand that each successive number name refers to a quantity that is one larger.



Students need practice writing numerals. Fill in the Line 0-9 allows students and opportunity for
repeated practice.

ESSENTIAL QUESTIONS

How can we use counting in our everyday life?
What is a numeral?
Why would we need to be able to read number words?
How can we record what we count?

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MATERIALS

ten sided die or 0-9 spinner
game board for each student
pencil

GROUPING



The object of the game is to be the first person to trace a complete line 0-9. To play the game,
player 1 will roll the die and trace the number the die lands on. He/she may choose any row to
trace the number. Next, Player 2 will roll the die and trace the number the die lands on. He/she
may choose any row to trace the number. If a number has already been traced in all 4 rows then
they player loses that turn. The first player to trace all the numbers in one row wins!


How many numbers do you need to win?
What numbers do you need to win?
Who is closer to winning? How do you know?
What number do you have the most of? Least of?

DIFFERENTIATION

Extension
Have students play Fill in the Line without the recording sheets. Have them record the
numerals in their math journal. Spatial recognition will be critical as students need to
leave space for the unwritten numerals.
Using a ten sided dice or spinner, students could record the numeral that is one less or 1
more than what was rolled. (Example: if a 3 was rolled, the student would record 2 or 4
depending on the rule)

Intervention
Correctly writing digits/numbers is an ongoing process that requires ongoing practice
throughout the year.
Students can practice writing numerals in the sand, with finger paint, or with a dry erase
marker on the desk.

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Fill the Line 0 to 9!

Fill the Line 0 to 9!

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_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

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Provide a paper clip or transparent spinner to use with the templates below. Place a pencil point inside one end of
the paper clip and hold with one hand. Use the other hand to flick the paperclip and it will spin. Students will need
to have practice with this prior to this activity. Great fine motor skill developer!

0-9 Spinner

nine zero 9 0
eight
one
8 1

seven two 2
7
six three
6 3
five four
5 4

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CONSTRUCTING TASK: The Rekenrek?
The Rekenrek can be used throughout the year and incorporated in a variety of tasks
to enforce concrete representation of numbers and strategies. Adapted from
www.k-5teachingresources.com





cardinality.
object.
were counted.

MCC.K.MD.3. Classify objects into given categories; count the numbers of objects in each




The Rekenrek is a math tool created by Adrian Treffers at the Freudenthal Institute in Holland.
Translated to English, Rekenrek means “counting rack”. The Rekenrek is composed of 20
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beads in two rows of ten with five red and five white on each rod. Although the Rekenrek may
look similar to an abacus, it differs because its structure is based around fives as opposed to tens.
The five-structure represents the five fingers on each of our hands and five toes on each of our
feet. Tournaki et al (2008) concluded that the structure of five utilized by the Rekenrek was extremely
helpful in the advancement of students’ number sense. In addition to increasing number sense,
Tournaki et al (2008) recognized that the Rekenrek acted as a facilitator of knowledge as
students develop efficient thinking strategies. Gravemeijer (1991) stated that materials
themselves cannot transmit knowledge to the learner, however it can make numbers and
relationships accessible to students to later obtain fact mastery and fluency. More information
on the Rekenrek can be found at http://www.mathlearningcenter.org/media/Rekenrek_0308.pdf.

Gravenmeijer, K. (1991). An Instruction-Theoretical Reflection On The Use Of Manipulatives.
Tournaki, N., Bae, Y., & Kerekes, J. (2008). Rekenrek: A manipulative used to teach addition
and subtraction to students with learning disabilities.

ESSENTIAL QUESTIONS

How can the Rekenrek help me as a mathematician?

MATERIALS

Cardboard
2 Pipe Cleaners or beading elastics
20 Beads (10 red/10 white)
Rekenrek Recording Sheet (optional)

GROUPING

The Rekenrek can be used whole group, small group, partner task, and individually


Making the Rekenrek:
Poke two small holes at each end of the cardboard, about 1in. in from the side.
Cut two 5in. lengths of elastic or use pipe cleaners. Place one end of each piece of elastic
into the holes at one end of the board and tie in a knot at the back or poke a pipe cleaner
through the hole and bend it.
Place five white beads and five red beads on each length of elastic or pipe cleaner.

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Once all the beads are on, thread the elastic through the holes on the other end and tie
securely so that the elastic is pulled tight, or put the other end of the pipe cleaner through
and bend.

○○○○○●●●●●

○○○○○●●●●●
Tasks:

Make It: teacher says or shows a number as students model the number using their
Rekenrek. It is important in this task for students to share the different ways they
modeled the number.

Flash It: teacher flashes a teacher made Rekenrek with a particular number and students
model what they saw. To extend student thinking, reduce the amount of time the teacher
Rekenrek is shown to students. The students could also have to model the number a
different way from the way that is flashed on the teacher Rekenrek.

Comment:

Rekenrek Norm Setting: When looking at the Rekenrek, the beads should be pushed over to the
‘Start Position’ (the right hand side), with the white beads farthest right and the red beads next to
them on the left. Note that the start position has the beads on the right so that when a student
pushes the beads over they can ‘read’ the quantity on the Rekenrek from left to right.

Students can record their thinking and modeling of the Rekenrek on the recording sheet. Once
students are familiar and comfortable drawing a pictorial representation with the recording sheet,
have them record/represent directly in their math journals.

Many of the activities with dot cards, Rekenreks and ten frames are interchangeable. The use of
multiple manipulatives to show number and quantity only reinforces a student’s understanding of
number which in turn increases number sense.


What number have you modeled?
What did you see? How do you know?
How many fives/tens do you see?
How many more do you need to make ten?
Which benchmark/anchor of 5/10 is your number closest to?
Can you build the number a different way?
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DIFFERENTIATION

Extension/Invention
Differentiating with the Rekenrek can be achieved through a variety of techniques which
are controlled by the teacher:
o The amount of time the Rekenrek is shown or flashed to students
o Increasing or decreasing the quantity made on the Rekenrek can help with
differentiating.

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My Rekenrek Recording Sheet Name:__________________

My number
sentence:_______________________________________________________

My number
sentence:_______________________________________________________

My number
sentence:_______________________________________________________

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PRACTICE TASK: Fill the Chutes
Approximately 1 Day then as Center (Van de Walle Activity 2.3)



cardinality.
object.
were counted.



BACKGROUND IKNOWLEDGE

Children will learn how to count (matching counting words with objects) before they understand
that the last count word indicates the amount of a set or the cardinality of a set. Children who
have made this connection are said to have the cardinality principle, which is a refinement of
their early ideas about quantity. (Van de Walle, 2006)

ESSENTIAL QUESTIONS


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MATERIALS:

20 counters per player
1 number cube (1-6)
Fill the Chutes game board

GROUPING:

1-4 players

TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION:

Part I
Place all the counters in a central pile where all players have access to them. Players take turns
rolling the dice and collecting/placing the corresponding amount of counters in their chute that
matches the roll the dice.

Both players must count out loud the total number of counters in their chute as they are added.

If a player has 3 spaces remaining unfilled and they roll 4 they cannot fill up the chute and have a
leftover or used counter. The chutes must be filled exactly.

Part II
After students have had an opportunity to engage in the activity the purpose of the roll alternates
each time.

Player 1 rolls and adds counters to the chute as they count out loud. On the next roll player 1
removes counters from the chute counting backwards. The first player to fill the chute wins the
game. This version of the activity helps with counting forward/backwards number sequence
starting a number other than 0 or 1.


What number did you roll?
How many counters do you have in your chute right now?
What number do you need roll to fill your chute?
Which chute has the most? Least?

DIFFERENTIATION

Extension
Change the value of each square to 10 and have students skip count by 10 to 100. Note:
the chute won’t be filled if students play to 100.After students are familiar with skip
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counting forward by tens, have them alternate rolls to skip count backwards and
forwards.

Intervention
Because the students must say the total number of counters out loud, the numerals for
each square could be written on the game board to help with number recognition and
counting forward and backwards..

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PRACTICE TASK: Race to 20
Approximately 1-2 days




cardinality.
d. When counting objects, say the number names in the standard order, pairing each object
object.
e. Understand that the last number name said tells the number of objects counted. The
were counted.
f. Understand that each successive number name refers to a quantity that is one larger.

(For descriptors of standard cluster, please see the Grade Level Overview)




Children will learn how to count (matching counting words with objects) before they understand
that the last count word indicates the amount of a set or the cardinality of a set. Children who
have made this connection are said to have the cardinality principle, which is a refinement of
their early ideas about quantity. (Van de Walle, 2006)

ESSENTIAL QUESTIONS

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MATERIALS:

2 different colored counters
1 number cube (1-6)
Race to 20 game board

GROUPING:

Partners (2 players)

TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION:

Each player places their counter in the starting square.
Players take turns using the spinner and move their counter the corresponding number of
spaces. Players must state what square they are on and count out loud in sequence to the
new square.
Player alternate turns with his/her partner until one player reaches 20.


As students are engaged in Race to 20, observe how students move the counter to locate the new
place on the game board. Is the student counting by ones, or are they using a strategy? If so,
which one?
How many squares do you need to win the game?
What square are you on now?
What is the number of the next square?

DIFFERENTIATION:

Extension
Each time a student rolls a die they alternate between counting spaces forwards and
backwards. The number of spaces is determined by the amount shown on the die.
Example: If player one rolled a six, they would move forward six spaces. If on the next
turn, player one rolled a three, they would move backwards three spaces. Students would
continue on until a player reached 20.

Intervention
Because the students must say the numerals aloud, the numerals could also be written on
the game board to assist with location identification and sequential counting.

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Start

5
0
Materials:
2 different colored counters
10
1 number cube (1-6) Race to 20
Directions:
-Each player places their counter in the starting square
numbered 0.
-Players take turns using the spinner and move their counter the
corresponding number of spaces. Players must state what square
they are on and count out loud in sequence (forward or backward)
to the new square.
-Players alternate turns until one player reaches 20.

Finish

15
20
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CONSTRUCTING TASK: The Cardinal Cup (0-10)
Approximately 1-2 days then repeated




cardinality.
object.
were counted.





Students should learn that counting objects in a different order does not alter the result, and they
may notice that the next whole number in the counting sequence is one more than the number
just named. Children should learn that the last number named represents the last object as well as
the total number of objects in the collection (NCTM Principles and Standards, 2012).

ESSENTIAL QUESTIONS
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Why do we need to be able to count forwards and backwards?

MATERIALS

Cardinal Cup playing mat
Playing cards from Numerals, Pictures, Words Task
(10) cubes or counters for the cup and 20 counters to keep score
Cup
Math journal to record numbers

GROUPING



Part I(counting forward)

Students use the task cards from Numbers, Pictures, Words and place them in a pile face down.
Player 1 turns over the top card and places that many cubes in the cup and counts out loud as
each cube is placed into the cup. Once Player 1 is finished counting, Player 2 removes the
contents from the cup and verifies that the correct number of cubes was placed in the cup by
placing the counters on the counting mat. (1-to-1 correspondence). If the player was correct in
counting out the cubes they receive 1 chip to be placed on their ten-frame. The first player to fill
up their ten frame wins.

As students place cubes on the number line ask questions that pertain to ordinal numbers and
positional words. What color is the 5th cube? What color is next to the 7th cube?

Part II (counting forward and/or backwards)

Students use the task cards from Numbers, Pictures, Words and place them in a pile face down.
Player 1 turns over the top card and places that many cubes in the cup and counts out loud as
each cube is placed into the cup. Once Player 1 has finished placing all the counters into the cup,
they turn over the next card from the pile and add/ remove cubes to/ from the cup to match the
second card. As player 1 adds/removes cubes from the cup they must count out loud forward or
backward in sequence, with the starting number being the quantity in the cup. The key is that
player 1 must mentally retain the number of cubes that were in the cup after the first card and
adjust the quantity in the cup without recounting the initial set of cubes. The new quantity must
match the number displayed on the second card.

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After Player 1 has made the necessary change to the cup, Player 2 dumps the cup out to verify
that the quantity in the cup matches the second card by using the counting mat. If the card and
quantity match, player 1 gets a chip to place on their ten frame. The first player to fill up their
ten frame wins.

Comment:
Students can record the numeral they counted in their journal for practice.
Ordinal numbers and understanding of positional words can be introduced /revisited
through teacher questioning. (Example: What was the second number you had to count?)
Only using two different colored cubes would allow students to count while creating a
pattern. (Example: 1st cube red, 2nd cube blue, 3rd cube red, etc…)


How many cubes are there in this set?
What color is the 5th cube? (ordinal numbers)
What color comes after the blue cube? (positional words)
If you created a pattern using red, blue, red, blue (ABAB) what color would the 7th cube
be?

DIFFERENTIATION

Extension
Ordinal numbers and understanding of positional words can be introduced /revisited
through teacher questioning.
Only using two different colored cubes would allow students to count while creating a
pattern. (Example: 1st cube red, 2nd cube blue, 3rd cube red, etc…)

Intervention
Repeated practice is the best intervention. To develop counting engage students in
almost any game or activity that involves counts and comparison.
Have students model the Cardinal Cup with their Rekenrek or ten-frame.

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CONSTRUCTING TASK: Order the Dice
Approximately 1 day




MCC.K.CC.4 Understand the relationship between numbers and quantities; connect counting to
cardinality.
object.
were counted.

MCC.K.MD.3.Classify objects into given categories; count the numbers of objects in each





This activity task is designed for number sequence and recognition. It will also help students
start at a number that may not be one and continue counting forward.

ESSENTIAL QUESTIONS

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How can we use counting in our everyday life?
Why is it important to know how to put things in number order?
How does putting things in order keep things organized?
Why do we need to be able to put things in order?

MATERIALS

Order the Dice game board
Dice

GROUPING

Partner task


Player 1 throws (5) six-sided dice at once. Dice are placed in order from smallest to largest. Example:
5,4,5,1,3 are rolled and place in order. 1,3,4,5,5. Player 1 receives 2 chips for having three numbers in
order. 2 chips are added to Player 1’s ten frame and they say the new total amount of chips. First player
to fill their two ten frames win the game.

Number of dice in Order 3 4 5
Points (chips collected) 1 2 3


What strategy did you use to help you put the numbers in order?

DIFFERENTIATION

Extension
Use number cubes that are not numbered 1-6 (4-9) or increase the quantity of dice used
from 5 to 10.

Intervention
Give students 5 dice and have them arrange the dice so that they are numbered 1-5.
Give students a set of cards from Numeral, Picture, Word (use only one form of card).
Shuffle up the cards and have the students practice putting them in order.

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Order the Dice
Player 1 throws (5) six-sided dice at once. Dice are place order from smallest to largest.
Example: 5,4,5,1,3 are rolled and place in order. 1,3,4,5,5. Player 1 receives 2 chips for
having three numbers in order. 2 chips are added to Player 1s ten frame and they say the
new total amount of chips. First player to fill their two ten frames win the game.
Number of Dice in Order 3 4 5
Points (chips collected) 1 2 3

Player 1 Scoreboard

Player 2 Scoreboard

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CONSTRUCTING TASK: More or Less
Approximately 1 day (Adapted from Van De Walle 2.1)




MCC.K.CC.3.Write numbers from 0 to 20. Represent a number of objects with a written

MCC.K.CC.4.Understand the relationship between numbers and quantities; connect counting to
cardinality.
number of objects is the same regardless of their arrangement or the order in which they were
counted.




The concept of “more, “less” and the “same” are basic relationships contributing to the overall
concept of number. Children begin to develop these ideas before they begin school. Children
entering kindergarten can almost always choose the set that is “more” if presented with sets that
are quite obviously different in number.

ESSENTIAL QUESTIONS

What is the difference between “more” and “less”?

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MATERIALS

Version 1 (Numbers 0-8)
Recording Sheet and game board
20 red/yellow counters
6 sided dice (1-6)
More/Less Spinner or Dice

Version 2 (Numbers 2-11)
Recording Sheet and game board
20 red/yellow counters
6 sided dice (4-9) (use wooden block)
More/Less Spinner or Dice

GROUPING



Comment
There are 2 version of this game. Each version can be played with more/less OR more/less 1&2.
The spinners provided can be used or dice/wooden blocks can be used to take place of the
spinners. The following description is generic for both games.

Player 1 rolls the dice (1-6) or (4-9) and spins the spinner (more/less or more/less 1&2). The
player covers the number in the way the dice relates to the spinner.

Part I (More/Less):
Example: if player 1 rolls a 5 then spin less, they can cover any number less than 5. (4,3,2,1, or
0) Watch the number the student covers as it relates to getting 3 in a row. Are they randomly
picking a number to cover? Or are they choosing the number to cover based on their best chance
to get 3 in a row?

Part II (More/Less 1&2):
Example: if you roll a 5 and spin 2 more, you count forward 2 from 5 to end at seven.

As students play, they record the number they rolled on the recording sheet. Then they record
what they spun (more/less, 1, more 1 less, etc….). Students then record what they covered on
the game board. They justify this in the “because” section by writing an equation or another
justification for covering. (Example: A player could say she rolled one more than 8. That’s 9,
because one more is the next number, so in the space she wrote “it’s next.”) First player to get 3
counters in a row wins.
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What does “more” mean? What does “less” mean?
What numbers do you need to win?
Why did you choose that number?
If you spun “2 more” what number would need to roll to win?

DIFFERENTIATION

Extension
Have the students model their actions using a ten-frame or Rekenrek. This will also help
students to record their actions.

Intervention
Allow the students to model with a ten frame or through the use of a number line.

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(0-8) More or Less -3 in a Row
Materials: Rules: Player 1 rolls the dice and spins the spinner. Player 1 covers a
spinner number square that relates to the spinner.
dice (Example: if player 1 rolls a 5 and spins “less”, they can cover any
counters number less than 5. (4, 3, 2, 1, 0). If player 1 spins “2 less “ they would
cover 3. Record what you did on the More or Less recording sheet.
First player to get 3 counters in a row wins.

7 8 6 0
4 5 4 2
3 3 6 5
1 0 2 4
5 7 1 8
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Math Unit 1

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